It sounds to me as if you are opening the group's wrapper and copying the geometry inside. Then you are pasting it while the group is still open for editing leaving that geometry inside the same wrapper. Instead you should make sure the group is closed. and copy it wrapper and all. When you select the group for copying it should have a blue box around it. That indicates you've got the whole group to copy.
Next, instead of copying with Ctrl-C you should be using the Move tool to copy the group. Select the group, get the Move tool and hit Ctrl to toggle on the copy function. Then click on the the group and move the copy in the desired direction. You can specify the distance by letting go of the mouse and typing the desired distance. You should check out the online help files (use the Help menu) on the Move tool.
If you are creating a stud wall, you can do something such as copy the first stud and type 16, hit Enter. Then immediately type *n or Xn where n is the number of copies you want in addition to the original. Each copy will be 16 in. from the previous one. It's dead simple to create a stud wall that way. As in real construction, it is likely the last one will need to be moved in a little bit but you can do that easily enough when you get tot hat point.
Also, instead of making the stud a group, make it a component. You can save it for later use and leverage a number of strengths components have over groups.
